By rotating this we can also turn off the light in the fifth row fourth column. In order to turn this one off, we will press the buttons as given in the following table. To do this, we will start at the bottom right of the board turning off lights reducing the number of lights on.įor the first case, we will start with the light in the fourth row and fifth column shown below. Upper middle Maximum lights Lower Right Our goal will now be to shift the lights over to upper left.
Our new maximum lights on is given below on the right. Again this will affect other lights, but none of the lights we turned off in the center or middle circle. In order to turn the corners off, we can just press the corner buttons. Note that this will change the lights in the first row second through fourth column, but it will not affect any of the lights we previously turned off. To begin with, suppose that we want to turn off the light in second row third column in the picture below we can press the button in the first row and third column. Corner Light Maximum lights Middle circle and Corners For the next lights, we will continue by reducing lights. Therefore, the maximum number of lights left on is given by the right picture. Through symmetry, we can do the same for the other lights in the corner of this middle circle. Therefore, if we start with the picture on the left below, we can turn that light off without affecting any other lights. The corresponding lights we will need to hit for this are given in the table below. This light can also be turned off without affecting any other light. In particular, we will focus on the light in the second row second column of the entire board. Center Light Max number of lights Middle Corner We now move out from the center to the surrounding lights. In order to do this, we would press the buttons as described in the table below. However, it will be helpful later if we describe a way to turn this off without changing any other light. The first piece we will turn off is the center piece. In doing this, we will be able to decrease the maximum number of lights on, hopefully, to 0 lights. We will then continue to turn lights off. We will then define an perform some action that will lead to turning a specific light off. Here, our goal will be to start with any board. We could, then, individually turn off any one light until all the lights were out. Last time, we were able to give a method for solving any puzzle by showing how to change only one light at a time.
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